Rietveld方法原理

nullnull 陈小龙 中国科学院物理研究所 2006.10 杭州Lecture notesRietveld 方法 快递客服问题件处理详细方法山木方法pdf计算方法pdf华与华方法下载八字理论方法下载 原理nullIsostructural CompoundsNd2CuO4 Gd2CuO4 a=0.39419nm a=3.8938nm c=1.21627nm c=11.8810nm ZNd=0.353 ZGd=0.349Nd(Gd)nullXRD patterns of Solid solution NaSr4-xBaxB3O9 (0≤x≤4) :First cubic borate only with BO3 Cubic borates are estimated below 1%.L. Wu, X.L. Chen, et. al. 2004Phase transition of BaTiO3 from tetragonal to cubic at about 132˚CPhase transition of BaTiO3 from tetragonal to cubic at about 132˚CYBa3B3O9: Phase transition and structure determinationYBa3B3O9: Phase transition and structure determinationS. G.: P63cm (No. 185) a=9.4235(4)Å, c=17.602(1) Å 1100C S. G.: R-3 (No. 148) a=13.0441(1)Å, c=9.5291(1) Å 1140C X.Z. Li, X.L. Chen, et. al. 2004LiAlB2O5: Search for new SHG materialsLiAlB2O5: Search for new SHG materials[B3O7]5- & [AlB2O7]5-null Structural Data for LiAlB2O5nullFinal Refinement of New compound of LiAlB2O5Structure vs Temperature:KCaCO3FStructure vs Temperature:KCaCO3F0.8120(3) 0.1880(3) 0.5 by x-ray datanullLiSr4B3O9 : A comparison between structure determination from single-crystal and powder X-ray diffractiona= 14.9470 Å S.G: Ia-3d (ZB/ZSr)2=(3/38)2 0.6%Single-crystal: Rint=0.0745 R1(all data)=0.0695 wR2(all data)=0.1887 with weighting scheme: W=1/[σ2(Fo2)+(0.0000P)2+359.71P] where P=(Fo2+2Fc2)/3 SDPD: RB=0.07 Rp=0.0609 Rwp=0.0811 Rexp=0.0314What is a Rietveld Refinement?What is a Rietveld Refinement? -a standard treatment of powder diffraction data to make the final structural model achieve the accepted criterion; -a best known method that fully makes use of the step-mode scanned data to dig out a lot of structural and other information; - a procedure for structural solution in nature. What can we get to perform a Rietveld refinement? What can we get to perform a Rietveld refinement? Lattice Parameters Quantitative phase Analysis Atomic Positions Grain size Atomic Occupancy Incommensurate Structure Debye Temperatures Structure factors Crystallinity Phase transitions Magnetic structures ……History ReviewHistory ReviewRietveld originally introduced the Profile Refinement method (Using step-scanned data rather than integrated Powder peak intensity) (1966,1967) Rietveld developed first computer Program for the analysis of neutron data for Fixed-wavelength diffractometers (1969) Malmos & Thomas first applied the Rietveld refinement method (RR) for analysis of x-ray powder data collected on a Ginier Hagg focusing Camera (1977) Khattack & Cox first applied the RR to x-ray powder data collected on a diffractometer (1977) Conference on Diffraction Profile Anlysis Sponsored by IUCr in Poland, suggested the term “Rietveld Method”(1978) Wiles and Yang developed a general computer program (D.B.W) for both x-ray & neutron diffraction data (fixed wavelength)(1981) Von Dreele, Jorgensen and Windsor extended to the program to the neutron diffraction data (1982) Fitch et al, 193 refined parameters,UO2 DAs.4D2O (1982)Aminoff Prize, Stockholm,1995Aminoff Prize, Stockholm,1995H.M. Rietveld Acta crystallogr., 22, 151 (1967). H.M.Riveted, J. Appl. Crystallogr., 2, 65 (1969). nullStructural model Raw dataRietveld RefinementRefined modelshiftHow RM works?How RM works?The RM refines a structure by minimizing a quantity through the Newton-Raphson algorithmwhere, yi is the observed intensity at a certain 2, yc,i is the calculated intensity at the same angle, wi is a weight, we usually take wi=1/yi i=1,2,…n =( 1 2 …p), the parameters to be refined.Given a solution =opt(1, 2… p) that approximately satisfy the above equation. To find a better solution, we begin an iterative process by expanding into a Taylor series.Given a solution =opt(1, 2… p) that approximately satisfy the above equation. To find a better solution, we begin an iterative process by expanding into a Taylor series.A1=bA1=b1= 0+ 11 is a shift. S is the scale factor of the phase  Lh contains the Lorentz, polarisation and multiplicity factors. Fh is the structure factor Ah is the absorption correction Ph is the preferred orientation function Ω is the reflection profile function that models both instrumental and sample effects S is the scale factor of the phase  Lh contains the Lorentz, polarisation and multiplicity factors. Fh is the structure factor Ah is the absorption correction Ph is the preferred orientation function Ω is the reflection profile function that models both instrumental and sample effects nullwhere, fi atomic scattering factor for ith atom xi, yi and zi the fractional coordinates for ith atomThe mean square displacement of the atom in a direction normal to the reflecting planesnullWhere RELAX is relaxtion factors that are used to control the shifts to avoid divergence; and CC is a multiplier.null What we need to perform a RR? A set of step-mode scanned data, usually 2=10-120˚ or more, step  2=0.02˚ collecting time is instrument dependent from 1-20s for laboratory diffractometer; An initial structural model having roughly accurate lattice constants, correct space group and approximate atomic positions nullHow we obtain an initial structural model? - solid solutions usually adopt same structure types of their parent compounds; NaSr4-xBaxB3O9 (0≤x≤4) Compounds with same chemical formula YBa2Cu3O7 and NdBa2Cu3O7 but always alert that exceptions are not uncommon La2CuO4 and Nd2CuO4 Try and error Ab inito structure determination Is the compound known? Crystallographic Structure DatabasesIs the compound known? Crystallographic Structure DatabasesICSD (Minerals and Inorganics) http://www.fiz-karlsruhe.de/ Minerals and Inorganic Over 60000 entries Cambridge Structure Data Bank） http://www.ccdc.cam.ac.uk Organics & Organometallics Over 250000 entriesICDD diffraction data http:http://www.icdd.com/ Inorganic & Organic Over 140000 entries NIST Crystal Data http://www.nist.gov/srd/nist3.htm Inorganic & Organic Over 230000 entriesnullA new structural database(2003): aimed at freely retrieving data18000 Patterns already!Parameters in PCR fileParameters in PCR fileThe parameters in PCR file can be divided into three categories -relating only to samples, refinable such as atomic positions, temperature factors -relating both to samples and instruments such as scale factors, FWHM (Full width at half maximum) -user-specified parameters such as BKPOS, Nba nullCodewords(I)Codewords(I) codewords are used to control parameters when to be refined, when to be fixed and when to be constrained and etc. A codeword is formed as C=S(10P+CC) Where S stands for the sign mark, P is an ordinal number set by users from 1 to p, the maximum number of parametersCodeword(II)Codeword(II)For example, an atom Ca position is (0,0,z) with z to be refined from its initial value 0.1. The codeword in your PCR file looks like the following …… Ca1 Ca+2 0.0 0.0 0.1 … 0.0 0.0 120.5 … …… Here, S=1, P=12, and CC=0.5. That means that z of Ca1 is the 12nd parameter to be refined in the iterative process, and x and y of Ca1 occupy special positions not needed to be refined. Codeword(III)Codeword(III)Another example: the lattice constants of a tetragonal compound are to be refined. The codeword in your PCR file looks as follows 3.891 3.891 11.732 51.0 51.0 61.0 In this case, constraint is put on a and b by using the same codeword since a=b always holds in tetragonal compoundsCodeword(IV)Codeword(IV) One more example: the occupancies of two kinds of atoms at one site are to be refined. Solid solutions are the most common among this kind of refinements. The codewords in your PCR file are set as Y Y+3 …… 0.8 …… 10.3 Yb Yb+3…… 0.2 …… -10.3 Only in this way are the occupancies guaranteed to satisfy Occ(Y)+Occ(Yb)=1Codeword(V)Codeword(V)each parameter usually controlled by one codeword. Be alert that one codeword should be given to two or more parameters that are irrelevant; - there is no limit to choose ordinal numbers. But we usually set the first ordinal numbers to global parameters such as zero point, background parameters and the etc. Modeling backgroundsModeling backgroundsThe background intensity bi at the ith step may be obtained by any of the following three method. a specified background function, usually a polynomial; linear interpolation between user-selected points in the pattern A user-supplied function Control flagsControl flagsThe choice of background type is indicated by a control flag Comment line(4) Job Npr Nph Nba Nex… 0 5 1 0 2 Nba: =0 Refine background with a polynomial =1 Read background from file COFHIL.bac =2,3,…,N linear interpolation between N given points … nullWhere Bm are parameter to be refined BKPOS is a user-specified parameter, origin of polynomial function, non-refinable. If 2=BKPOS, we see bi=B0 Users can look into their data files to set the values of BKPOSProfile functions (I)Profile functions (I)Gaussian (G) Npr=0 Lorentzian (L) Npr=1Parameter to be refined: Hk, Full Width at Half Maximum (FWHM)Hk=0.2Hk=0.2Profile functions (II)Profile functions (II)Mod.I Lorentzian Npr=2 Parameter to be refined: Hk, Mod.I Lorentzian Npr=3 Profile functions (III)Profile functions (III)Psudo-Voigt Npr=5Parameters to be refined: Hk, η0, Xη0 = shapePseudo-Voigt functions Hk=0.2Pseudo-Voigt functions Hk=0.2Profile functions (III)Profile functions (III)Pearson VIIParameters to be refined: Hk, m0, X,YProfile functions (IV)Profile functions (IV) (Mod-TCHZ pV) L(x) and G(x) have different FWHMs HL and HGParameters to be refined: HG and HLFull width at half Maximum (FWHM) Full width at half Maximum (FWHM) For Npr=0…6, Hk=HG For Npr=7, HL is required apart from HG Typical variations of FWHM vs 2 Typical variations of FWHM vs 2 Summary for the parameters to be refined with different profilesSummary for the parameters to be refined with different profilesNpr=0, Gaussian: U, V, W, Ig 3 Npr=5, pv: U,V,W,Ig, η0(Shape), X 5 NPr=6, Pearson VII: U,V,W,Ig, η0(Shape), X,Y 6 NPr=7, TCHZpv: U,V,W,Ig, X,Y,Sz 6nullPreferred orientations (I)Preferred orientations (I)Nor=0, Rietveld-Toraya ModelG1 and G2 are refinable parameters H is the acute angle between d*H and the normal to the crystallites (platy habit)Note: preferred orientation vector Pr1,Pr2 and Pr3 is needed to specify a priori by usersPreferred orientations (II)Preferred orientations (II)Nor=1, modified March’s ModelG1 and G2 are refinable parametersG1<1, platy habit, G1=1, no preferred orientation G1>1 Needle-like habitnullSystematic line-shiftSystematic line-shiftBragg-Brentano Geometry Specimen displacement Specimen Transparency SYCOSSYSIN : the linear absorption coefficient of the sampleWDT·FWHMWDT·FWHMWDT>5, preferably 10Monochromator polarization correctMonochromator polarization correct Incident angle to a monochromator CTHM=cos22=0.8009 for a graphite monochromator, CuK Asymmetry correction for profilesAsymmetry correction for profilesP1, P2, P3 , and P4 are parameters to be refinedAsymLim: peaks below this 2 angle limit are corrected for asymmetry nullOccupancyOccupancym is the site multiplicity, M is the multiplicity of the general site for a given space group. For example, KCaFCO3, P-6m2(187) K+1 occupies 1(a) site; O-2 3(k) site; general site 12(o), Occ(K+)=1/12=0.08333, Occ(O-2)=0.25, both atoms’ chemical occupancy=1.0 Agreement Factors (I)Agreement Factors (I)Profile FactorWeighted Profile FactorExpected Weighted Profile FactorAgreement Factors (II)Agreement Factors (II)Goodness of fit indicatorBragg FactorCrystallographic RF factorVariations of agreement factors and esd.Variations of agreement factors and esd.Hill & Madsen, Powder Diffraction(1987)nullAn estimation of S for an ideal refinementtake S≈1 since N>>PnullGaussianLorentzianR.J. Hill and H.D. Flack, J.Appl. Cryst. 20 (1987) 356-361Durbin-Watson statistic parametersDurbin-Watson statistic parametersd < QD : positive serial correlation QD< d <4-QD: no serial correlation d > 4-QD: negative serial correlationVariations of d, eds, Rwp and RB vs cyclesVariations of d, eds, Rwp and RB vs cyclesR.J. Hill and H.D. Flack, J.Appl. Cryst. 20 (1987) 356-361Comments on agreement factorsComments on agreement factorsRF and RB are more indicative of structural model fits Rp and Rexp are more indicative of overall profile fits RF ,RB , Rp and Rexp are not good indices for the refinements of different patterns S should be as close as 1 d is a more sensitive index over RF ,RB , Rp and Rexp At least Rp, Rwp and Rexp should be given when submitting a paper to a journalQuantitative Phase AnalysisQuantitative Phase Analysiswhere, Wj is the weight fraction for the jth phase; Sj is scale factor for the jth the phase; Zj is the number formula units per cell for the jth phase; Mj is the mass of the formula unit; Vj is the unit cell volume; tj Brindley coefficient that comes into effect when the linear absorption coefficients of phases in powder differ a lot to each other. nullKCaFCO3CaCO3nullMultiphase Rietveld AnalysisnullResult in *.out fileNo absorption correction is appliednullTo obtain a satisfactory quantitative phase analysis based on the Rietveld method, we should be cautioned: - Sample should be carefully prepared: powder is homogeneous in compositions and have a sufficient number of grains with random orientations; Structures of phases are well known; Absorption correction is applied whenever the phases differ a lot in their linear absorption coefficients. The Brindley coefficients can be consulted in the Fullprof Manual. Calculation of bond length, angle and bond valence sumCalculation of bond length, angle and bond valence sumESDsESDsnullBond valence sum is a good indicator of the structural validity. For details, see I.D.Brown, Acta Crystallogr. B48, 141(1992)EPS and Relax factors EPS and Relax factors Forced termination when shifts10000 Too less sample. Sample should fully cover the sample holder window; Overflow in low angle region; Too less angle region 2>120˚; Too large EPS that leads to false minimum; Improper profile function; Too less WDT values .Error messages (I)Error messages (I)‘Hole in Matrix’ : the number of parameters to be refined NPR larger than the number of codewords For example, you set NPR= 12, while one codeword 80.5 is missing or 101.0 is mistyped as 11.0 - ‘Negative FWHM’: HG2 <0, meaningless! Increasing the negative U,V, or W while set smaller Relax values in the ensuing the refinementsError messages (II)Error messages (II)- ‘No scattering factor’ : atom identifier ‘TYP’ is not recognizable by Fullprof. For example, Ca+2 is accepted while Ca2+ is not accepted. ‘Too many reflections’: For a given point, there are too many reflections contributing to the intensity that are beyond the software’s capacity. Usually this results from the false FWHM ‘Invalid integer’ or ‘Invalid real’: Examine the format of parameters Note: The software does not always give the correct lines where errors occur. Look into the nearby lines!nullchenhongMAC MXP18A-HFFormCon2.0nullhuming理学 DMAX 2000nullRefine scale factor S nullRwp=962 138, S=0.5E-2 0.64E-3nullRefine zero point along with SnullRwp=62.8 ZP≈0.05nullRefine background along with S and ZPnullRwp=32.9%Refine lattice parameters along with othersRefine lattice parameters along with othersnullRwp=28.9%nullRefine peak profile along with other parametersnull Rwp=18.9%Refine asymmetry Refine asymmetry null Rwp=16.8%nullRefine atomic coordinates: first two atoms Pb and S the number of parameters to be refined:22nullRwp=14.0%Further refine atomic coordinates of 3 O atoms Rwp=13.1%nullRefine temperature factors along other parametersnullRwp=12.5%nullRp=8.92%, Rwp=10.8%, Rexp=6.57%nullCalculate the bond lengths and bond valences And the results are stored in *.disnullababcPbSO4Fourier synthesisFourier synthesisSet Fou=4 in your Pcr filePbSO4Rp=7.3%-16.6% 5.82% 8.91% Rwp=8.2-20.0% 7.83% 10.8% Rexp=1.5%-7.0% 4.83% 6.71% GodF=1.3-7.4 1.6 1.6 Rp=7.3%-16.6% 5.82% 8.91% Rwp=8.2-20.0% 7.83% 10.8% Rexp=1.5%-7.0% 4.83% 6.71% GodF=1.3-7.4 1.6 1.6 Compare with the Rietveld Refinement Round RobinTotal 23 respondentsBackground excludednull Range mean single crystal this work a(Å) 8.4764-8.4859 8.4804(4) 8.482(2) 8.4818(1) b(Å) 5.3962-5.4024 5.3989(3) 5.398(2) 5.3997(1) c(Å) 6.9568-6.9650 6.9605(4) 6.959(2) 6.9614(1)Compare with the Rietveld Refinement Round RobinR.J.Hill, J. Appl.Cryst. 25, 589(1992)null Range mean single crystal this work Pb x 0.1875-0.1883 0.18783(4) 0.1879(1) 0.18785(7) z 0.1669-0.1683 0.16752(9) 0.1667(1) 0.16742(10) S x 0.0621-0.0673 0.0642(2) 0.0633(6) 0.0638(4) z 0.6799-0.6860 0.6838(4) 0.6842(7) 0.6834(6) O1x 0.902-0.924 0.9083(13) 0.908(2) 0.9069(11) z 0.585-0.601 0.5945(7) 0.596(3) 0.5929(14)Compare with the Rietveld Refinement Round Robinnull Range mean single crystal this work O2 x 0.177-0.200 0.1850(11) 0.194(2) 0.1894(11) z 0.523-0.548 0.5398(13) 0.543(2) 0.5423(14) O3 x 0.071-0.080 0.0778(5) 0.082(1) 0.0789(6) y 0.018-0.041 0.026(13) 0.026(2) 0.0214(9) z 0.806-0.819 0.8139(7) 0.809(2)